Conflict-free coloring of graphs

TL;DR

This paper investigates the maximum and typical conflict-free chromatic number of graphs, providing new bounds and asymptotics for random graphs, and relates it closely to the domination number.

Contribution

It resolves a key open question about the maximum conflict-free chromatic number of n-vertex graphs using a randomized construction and analyzes its evolution in Erdős-Rényi graphs.

Findings

Maximum conflict-free chromatic number for n-vertex graphs determined

Asymptotic behavior of conflict-free chromatic number in G(n,p) for p=omega(1/n)

For p ≥ 1/2, conflict-free chromatic number closely approximates the domination number

Abstract

We study the conflict-free chromatic number chi_{CF} of graphs from extremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erd\H{o}s-R\'enyi random graph G(n,p) and give the asymptotics for p=omega(1/n). We also show that for p \geq 1/2 the conflict-free chromatic number differs from the domination number by at most 3.